Problem: A group of adults and kids went to see a movie. Tickets cost $$5.00$ each for adults and $$3.50$ each for kids, and the group paid $$34.50$ in total. There were $5$ fewer adults than kids in the group. Find the number of adults and kids in the group.
Solution: Let $x$ equal the number of adults and $y$ equal the number of kids. The system of equations is then: ${5x+3.5y = 34.5}$ ${x = y-5}$ Solve for $x$ and $y$ using substitution. Since $x$ has already been solved for, substitute ${y-5}$ for $x$ in the first equation. ${5}{(y-5)}{+ 3.5y = 34.5}$ Simplify and solve for $y$ $ 5y-25 + 3.5y = 34.5 $ $ 8.5y-25 = 34.5 $ $ 8.5y = 59.5 $ $ y = \dfrac{59.5}{8.5} $ ${y = 7}$ Now that you know ${y = 7}$ , plug it back into ${x = y-5}$ to find $x$ ${x = }{(7)}{ - 5}$ ${x = 2}$ You can also plug ${y = 7}$ into ${5x+3.5y = 34.5}$ and get the same answer for $x$ ${5x + 3.5}{(7)}{= 34.5}$ ${x = 2}$ There were $2$ adults and $7$ kids.